On the map, "Wilderland", the Forest Gate is at about the same latitude as Thranduil's Caverns (on the eastern edge of Mirkwood). It lies about halfway between the point where the Grey Mountains abut Mirkwood in the north, and where the Carrock lies in the Anduin/Langflood to the south.

As soon as they [Thorin Oakenshield's company] left his [Beorn's] high hedges at the east of his fenced lands they turned north and then bore to the north-west. ... North of the Carrock the edge of Mirkwood drew closer to the borders of the Great River [The Anduin/Langflood], and though here the [Misty] Mountains too drew down nearer, Beorn advised them to take this way; for at a place a few days' ride due north of the Carrock was the gate [The Forest Gate] of a little-known pathway [The Elf-path] through Mirkwood that led almost straight towards the Lonely Mountain.

The Hobbit, Ch 7, Queer Lodgings

So now there was nothing left to do but to fill their water-skins at a clear spring they found close to the forest-gate, and unpack the ponies.

The Hobbit, Ch 7, Queer Lodgings

He [Bilbo] had gone just inside the forest [Mirkwood] after breakfast (a very poor one), and it had seemed as dark in there in the morning as at night, and very secret: "a sort of watching and waiting feeling," he said to himself.

The Hobbit, Ch 7, Queer Lodgings

They [Thorin Oakenshield's company] walked in single file. The entrance to the path was like a sort of arch leading into a gloomy tunnel made by two great trees that leant together, too old and strangled with ivy and hung with lichen to bear more than a few blackened leaves. The path itself was narrow and wound in and out among the trunks. Soon the light at the gate was like a little bright hole far behind, and the quiet was so deep that their feet seemed to thump along while all the trees leaned over them and listened.